Visual Reasoning in Euclid’s Geometry: An Epistemology of Diagrams

Abstract

It is widely held that the role of diagrams in mathematical arguments is merely heuristic, or involves a dubious appeal to a postulated faculty of “intuition”. To many these have seemed to exhaust the available alternatives, and worries about the status of intuition have in turn motivated the dismissal of diagrams. Thus, on a standard interpretation, an important goal of 19th Century mathematics was to supersede appeals to intuition as a ground for knowledge, with Euclid’s geometry—in which diagrams are ubiquitous—an important target. On this interpretation, Euclid’s presentation is insufficient to justify belief or confer knowledge in Euclidean geometry. It was only with the work of Hilbert that a fully rigorous presentation of Euclidean geometry became possible, and such a presentation makes no nonredundant use of diagrams. My thesis challenges these claims. Against the “heuristic” view, it argues that diagrams can be of genuine epistemic value, and it specifically explores the epistemology of diagrams in Euclid’s geometry. Against the “intuitive” view, it claims that this epistemology need make no appeal to a faculty of intuition. It describes in detail how reasoning with diagrams in Euclid’s geometry can be sufficient to justify belief and confer knowledge. And it shows how the background dialectic, by assuming that the “heuristic” or “intuitive” views above are exhaustive, ignores the availability of this further alternative. By using a detailed case study of mathematical reasoning, it argues for the importance of the epistemology of diagrams itself as a fruitful area of philosophical research.Philosophy of Mathematics, Epistemology & Methodolog